The moment I realized the difference was when working on a new monitor with 144hz. When connecting object with joints (e.g., building chains with revolute joint), they behave like smooth strings/chains and are completely stable. The same project 60hz behaves similar for low forces, but can easily break at obstacles (get disjoint for a short time) when applying higher forces. This is definitely fixable by preventing these forces and write additional code around it, but this in turn changes the behavior on the faster refresh rates to feel a little bit 'overtuned'.
Increasing the iterations helps to reduce the differences, but the lower latency still had a superior result. Also, the higher accuracy affects the performance on devices with higher refresh rates, even if they would not need it. The problem is that devices with fast refresh rates are not neccessarily better - they just have a faster display. This means that the best solution would probably be to adapt the accuracy according to the framerate, to find a tradeoff between CPU usage and 'reliable physics'.
Some devices (like some laptops) start getting ridiculously high refresh rates like 360hz, where these smaller differences in physics behavior might build up. This unfortunately means that one has not only to consider the weaker devices, but also the more powerful ones. For these it would propbably be the best to just cap the updates to a level that is considered 'save'.
However, I can definitely see the problem with independent clocks, especially if the system can't keep up with the updates. That said, I realized that often devices are powerful enough to manage the higher update rate - they just wont do it because of their display (especially when targetting stronger/newer devices). Especially when targetting PC and laptops, the refresh rate is probably not really tied to the specs of the CPU. This leads to the undesirable situation that weaker devices might ask for high framerates, while much stronger devices are held back by their display.
I agree that the update rate should never be set anywhere near the limit, but I would think that the cap introduced by the refresh rate is not the best option for all cases anymore.